Finite Math Examples

$\log_{3} (2 x - 3) = 2 \log_{3} (3) + \log_{3} (3 x - 2)$

Step 1

Simplify the right side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Logarithm base $3$ of $3$ is $1$ .

$\log_{3} (2 x - 3) = 2 \cdot 1 + \log_{3} (3 x - 2)$

Step 1.1.2

Multiply $2$ by $1$ .

$\log_{3} (2 x - 3) = 2 + \log_{3} (3 x - 2)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$\log_{3} (2 x - 3) - \log_{3} (3 x - 2) = 2$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{3} (\frac{2 x - 3}{3 x - 2}) = 2$

Step 4

Rewrite $\log_{3} (\frac{2 x - 3}{3 x - 2}) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{2} = \frac{2 x - 3}{3 x - 2}$

Step 5

Cross multiply to remove the fraction.

$2 x - 3 = 3^{2} (3 x - 2)$

Step 6

Simplify $3^{2} (3 x - 2)$ .

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Step 6.1

Raise $3$ to the power of $2$ .

$2 x - 3 = 9 (3 x - 2)$

Step 6.2

Apply the distributive property.

$2 x - 3 = 9 (3 x) + 9 \cdot - 2$

Step 6.3

Multiply.

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Step 6.3.1

Multiply $3$ by $9$ .

$2 x - 3 = 27 x + 9 \cdot - 2$

Step 6.3.2

Multiply $9$ by $- 2$ .

$2 x - 3 = 27 x - 18$

Step 7

Move all terms containing $x$ to the left side of the equation.

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Step 7.1

Subtract $27 x$ from both sides of the equation.

$2 x - 3 - 27 x = - 18$

Step 7.2

Subtract $27 x$ from $2 x$ .

$- 25 x - 3 = - 18$

Step 8

Move all terms not containing $x$ to the right side of the equation.

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Step 8.1

Add $3$ to both sides of the equation.

$- 25 x = - 18 + 3$

Step 8.2

Add $- 18$ and $3$ .

$- 25 x = - 15$

Step 9

Divide each term in $- 25 x = - 15$ by $- 25$ and simplify.

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Step 9.1

Divide each term in $- 25 x = - 15$ by $- 25$ .

$\frac{- 25 x}{- 25} = \frac{- 15}{- 25}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 25$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 25 x}{- 25} = \frac{- 15}{- 25}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 15}{- 25}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Cancel the common factor of $- 15$ and $- 25$ .

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Step 9.3.1.1

Factor $- 5$ out of $- 15$ .

$x = \frac{- 5 (3)}{- 25}$

Step 9.3.1.2

Cancel the common factors.

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Step 9.3.1.2.1

Factor $- 5$ out of $- 25$ .

$x = \frac{- 5 \cdot 3}{- 5 \cdot 5}$

Step 9.3.1.2.2

Cancel the common factor.

$x = \frac{- 5 \cdot 3}{- 5 \cdot 5}$

Step 9.3.1.2.3

Rewrite the expression.

$x = \frac{3}{5}$

Step 10

Exclude the solutions that do not make $\log_{3} (2 x - 3) = 2 \log_{3} (3) + \log_{3} (3 x - 2)$ true.

$No solution$